What exactly are the implications? Can a 32 bit linux run applications that are compiled as 64 bit? Or vice versa?

I just got a new machine with an i5 processor, I installed a copy of Ubuntu 10.10 which seems to be i686 which I new realize is 32 bit, but I think I have some 64 bit apps installed. Can this be right?

You MIGHT be able to run 32bit applications. It depends. Ubuntu is really good at including 32bit libs so I've never had a problem. But something like arch, getting 32 bit shared libraries on a 64 bit system isn't trivial. However if the binary doesn't have dynamic library dependences you should be fine. Unless you're building it from source.
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FalmarriJan 26 '11 at 18:13